1
vote
1answer
41 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
0
votes
1answer
26 views

Weakly sequentially continuous operators in Hilbert space are norm continuous.

Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous. I feel that this statement ...
1
vote
1answer
38 views

Weak convergence plus strong convergence

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to ...
2
votes
1answer
32 views

Weak convergence and infinite sum

Suppose that $\psi_n$ converges weakly to $\psi$ in a Hilbert space $H$. Assume further $\{\phi_k\}$ is an orthonormal sequence in $H$. Is it plausible that ...
0
votes
0answers
30 views

There is a unit sequence weakly converging to every element of unit ball

Suppose the Hilbert space $H$ has a countable (I assume Hilbert?) basis. Let $x \in H$ be such that $\lvert x \rvert \leq 1$. Show that there exists a sequence $\{u_{n}\}$ in $H$ with ...
2
votes
2answers
56 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
1
vote
1answer
89 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
2
votes
1answer
45 views

Weak and Norm convergence in Banach Space

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$. I'm trying to find a counterexample in a Banach ...
1
vote
1answer
41 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
1
vote
0answers
25 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
2
votes
1answer
79 views

Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert ...
1
vote
0answers
59 views

about a theorem of weakly lower semicontinuous functions

I am studying the proof of the following theorem Theorem: Let $E$ a Hilbert space and suppose that $\varphi :E \rightarrow R$ is a weakly lower semicontinuous functional. Suppose that $\varphi$ is ...
0
votes
1answer
89 views

Weak limits and subsequences

Let $S:X \to X$ be a (nonlinear) map between a Hilbert space $X$. I want to show that $S$ is weakly continuous, so if $x_n \rightharpoonup x$, then $S(x_n) \rightharpoonup S(x)$. To do this, I have ...
0
votes
1answer
14 views

If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $\limsup_{n \to \infty}(Au_n, u_n) \leq (b, u)$ if $A$ is continuous?

Let $H$ be a Hilbert space and let $A:H \to H$ be a (nonlinear) continuous map. If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $$\limsup_{n \to \infty}(Au_n, u_n) \leq ...
1
vote
0answers
48 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in ...
2
votes
2answers
268 views

Weak convergence in a Hilbert Space

What does it mean for a sequence $\{f_n\}_{n=1}^\infty\subseteq H$ to converge weakly? I know it means that it converges in the weak topology and I've read a few definitions of weak topology which all ...
2
votes
1answer
47 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
1
vote
1answer
82 views

Weak convergence equal to coordinate-wise convergence

Show for the Hilbert Space $\ell^2(\mathbb{N})$ weak convergence of a bounded sequence $(x_k)_{k \in \mathbb{N}}$ is equal to coordinate-wise convergence. A sequence $(x_k)$ is weak ...
1
vote
0answers
34 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
1
vote
1answer
331 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
5
votes
2answers
1k views

Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions. A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is ...
1
vote
1answer
43 views

Closed subspace of Hilbertspace

Let $X$ be a norm closed subspace of a Hilbert space $\mathcal H$. Is it true that if $x_n \in X$ converges weakly to $x \in \mathcal H$, then also $x \in X$ ?
3
votes
3answers
204 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have ...
0
votes
1answer
238 views

A problem about the strong convergence in a Hilbert space.

Let $X_n$ be a sequence in a Hilbert space H. If $\| X_n \| \to \| x \|$, show that $X_n$ converges strongly to $x$. Note that I'm not saying anything about weak convergence.
1
vote
2answers
141 views

On the uniqueness of certain weak cluster point in Hilbert space

The problem goes as follows: Let $\mathcal{H}$ be real Hilbert space, $C\subset \mathcal{H}$ be a subset. Let $\{x_n\}\subset\mathcal{H}$ satisfies the following property: ...
4
votes
1answer
247 views

Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
4
votes
2answers
160 views

about weak convergence in $L^{2}(0,T;H)$

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
1
vote
2answers
108 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
1
vote
1answer
65 views

Weak convergence in $L^2$ and CDF

Assume that for sequence $X_n \in L^2(\Omega,F,P)$ which converges in distribution to CDF $F_X$ ($F_n(t)\rightarrow F_X(t)$ for every point of continuity of $F_X$), we have also that $X_n$ converges ...
3
votes
1answer
786 views

Bounded sequence in Hilbert space contains weak convergent subsequence

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence. Is there any short proof? Thanks a lot.
2
votes
1answer
188 views

Weak convergence in Hilbert space L2 implies convergence in distribution?

Does weak convergence in $L^2$ (for $X_n, X \in L^2$ we say that $X_n$ converges weakly to $X$ ($X_n \rightarrow^w X$) if for every $Y\in L^2$ we have $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
2
votes
1answer
82 views

Limit of a sequence in the space $\ell_2$

I have difficulties in the following problem. Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that $$ u^{k+1}=(1-\alpha)u^k+\alpha ...
1
vote
2answers
82 views

Sequence of operators in a Hilbert space

The question is: Let $H$ be a Hilbert space and $\{T_n\}$ be a sequence in $B(H)$ such that $\lim_{n\rightarrow\infty}\langle x, T_n y \rangle = 0$ for all $x, y \in H$. Prove or disprove $\sup_n ...
1
vote
0answers
31 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
1
vote
2answers
198 views

Weak convergence-exercice

Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$ Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
6
votes
0answers
396 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
1
vote
1answer
115 views

Weak to strong mapping

Let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be strongly monotone if there exists $\alpha>0$ such that $$ \langle F(u)-F(v), u-v\rangle\geq \alpha \|u-v\|^2, \quad ...
2
votes
1answer
108 views

Weak and strong convergence of sequence of linear functionals

Is this sequence of linear functionals weakly (strongly) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_2\,?$$
2
votes
3answers
753 views

Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence ...